Bifurcation of periodic solutions of a system of differential equations
A bifurcation problem of periodic solutions to the differential equations of the form (1) \(\dot x=f(x,\lambda)x\) where \(x\in E_ n\), \(f(x,\lambda)\) is an \(n\times n\)-matrix, \(\lambda \in E_ m\), \(\lambda\) is a parameter and \(E_ s\) is an s-dimensional vector space is investigated. A case when the first order approximation matrix of the system (1) has two complex-conjugate eigenvalues is discussed. Additional conditions (on the velocities of intersection of the imaginary axis by the complex-imaginary eigenvalues) which provide an existence of the bifurcation of the periodic solutions of the system (1) are found. The system of differential equations \(\dot x=C(\lambda)x+L(x,\lambda)x,\) where \(x=(x_ 1,x_ 2)\), \(C(\lambda)\) and \(L(x,\lambda)\) are \(2\times 2\)-matrices and \(\lambda\) is a scalar parameter, is considered as an example.
scientific article; zbMATH DE number 4144420